Download work of a force

THE WORK OF A FORCE, THE PRINCIPLE OF WORK AND ENERGY & SYSTEMS OF
PARTICLES
1. Calculate the work of a force.
2. Apply the principle of work and
energy to a particle or system
of particles.
APPLICATIONS
A roller coaster makes use of gravitational forces to assist the cars in reaching high
speeds in the “valleys” of the track.
How can we design the track (e.g., the height, h, and the radius of curvature, r) to
control the forces experienced by the passengers?
APPLICATIONS
(continued)
Crash barrels are often used along
roadways in front of barriers for crash
protection.
The barrels absorb the car’s kinetic energy
by deforming.
If we know the velocity of an
oncoming car and the amount of
energy that can be absorbed by each
barrel, how can we design a crash
cushion?
WORK AND ENERGY
Another equation for working kinetics problems involving particles can be derived
by integrating the equation of motion (F = ma) with respect to displacement.
By substituting at = v (dv/ds) into Ft = mat, the
result is integrated to yield an equation
known as the principle of work and energy.
This principle is useful for solving problems that involve force, velocity, and
displacement. It can also be used to explore the concept of power.
To use this principle, we must first understand how to calculate the work of a
force.
WORK OF A FORCE (Section 14.1)
A force does work on a particle when the particle undergoes a displacement along
the line of action of the force.
Work is defined as the product of force and
displacement components acting in the same
direction. So, if the angle between the force and
displacement vector is q, the increment of work dU
done by the force is
dU = F ds cos q
By using the definition of the dot product and
integrating, the total work can be written as
r2
U1-2 =

r1
F • dr
WORK OF A FORCE (continued)
If F is a function of position (a common case) this
becomes
s2
U1-2
=
 F cos q ds
s1
If both F and q are constant (F = Fc), this equation further simplifies to
U1-2 = Fc cos q (s2 - s1)
Work is positive if the force and the movement are in the same direction. If
they are opposing, then the work is negative. If the force and the
displacement directions are perpendicular, the work is zero.
WORK OF A WEIGHT
The work done by the gravitational force acting on a particle (or weight of an
object) can be calculated by using
y2
U1-2 =
 - W dy
y1
U1-2 = - W (y2 − y1) = - W Dy
The work of a weight is the product of the magnitude of the particle’s
weight and its vertical displacement. If Dy is upward, the work is negative
since the weight force always acts downward.
WORK OF A SPRING FORCE
When stretched, a linear elastic spring develops a force of
magnitude Fs = ks, where k is the spring stiffness and s is
the displacement from the unstretched position.
The work of the spring force moving from position s1 to position s2 is
s2
s2
U1-2 =
s F ds
s
1
=
 k s ds
= 0.5 k (s2)2 – 0.5 k (s1)2
s1
If a particle is attached to the spring, the force Fs exerted on the particle is opposite to
that exerted on the spring. Thus, the work done on the particle by the spring force will
be negative or
U1-2 = – [ 0.5 k (s2)2 – 0.5 k (s1)2 ] .
SPRING FORCES
It is important to note the following about spring forces.
1.
The equations above are for linear springs only! Recall that a linear spring
develops a force according to
F = ks (essentially the equation of a line).
2.
The work of a spring is not just spring force times distance at some point, i.e.,
(ksi)(si). Beware, this is a trap that students often fall into!
3.
Always double check the sign of the spring work after calculating it. It is positive
work if the force on the object by the spring and the movement are in the same
direction.
PRINCIPLE OF WORK AND ENERGY
(Section 14.2 & Section 14.3)
By integrating the equation of motion,  Ft = mat = mv(dv/ds), the
principle of work and energy can be written as
 U1-2 = 0.5 m (v2)2 – 0.5 m (v1)2 or T1 +  U1-2 = T2
U1-2 is the work done by all the forces acting on the particle as it
moves from point 1 to point 2. Work can be either a positive or
negative scalar.
T1 and T2 are the kinetic energies of the particle at the initial and final
position, respectively. Thus, T1 = 0.5 m (v1)2 and T2 = 0.5 m (v2)2. The
kinetic energy is always a positive scalar (velocity is squared!).
So, the particle’s initial kinetic energy plus the work done by all the
forces acting on the particle as it moves from its initial to final position
is equal to the particle’s final kinetic energy.
PRINCIPLE OF WORK AND ENERGY (continued)
Note that the principle of work and energy (T1 +  U1-2 = T2) is not a vector
equation! Each term results in a scalar value.
Both kinetic energy and work have the same units, that of energy! In the SI system,
the unit for energy is called a joule (J), where 1 J = 1 N·m. In the FPS system, units are
ft·lb.
The principle of work and energy cannot be used, in general, to determine forces
directed normal to the path, since these forces do no work.
The principle of work and energy can also be applied to a system of particles by
summing the kinetic energies of all particles in the system and the work due to all
forces acting on the system.
WORK OF FRICTION CAUSED BY SLIDING
The case of a body sliding over a rough surface merits special consideration.
Consider a block which is moving over a rough surface.
If the applied force P just balances the resultant
frictional force k N, a constant velocity v would be
maintained.
The principle of work and energy would be applied as
0.5m (v)2 + P s – (k N) s = 0.5m (v)2
This equation is satisfied if P = k N. However, we know from experience that friction
generates heat, a form of energy that does not seem to be accounted for in this
equation. It can be shown that the work term (k N)s represents both the external work
of the friction force and the internal work that is converted into heat.
EXAMPLE
Given:
When s = 0.6 m, the spring is not
stretched or compressed, and the 10
kg block, which is subjected to a force
of 100 N, has a speed of 5 m/s down
the smooth plane.
Find:
The distance s when the block stops.
Plan:
Since this problem involves forces, velocity and displacement, apply the
principle of work and energy to determine s.
EXAMPLE (continued)
Solution:
Apply the principle of work and energy between position 1
(s1 = 0.6 m) and position 2 (s2). Note that the normal force (N) does no work since it
is always perpendicular to the displacement.
T1 + U1-2 = T2
There is work done by three different forces;
1) work of a the force F =100 N;
UF = 100 (s2− s1) = 100 (s2 − 0.6)
2) work of the block weight;
UW = 10 (9.81) (s2− s1) sin 30 = 49.05 (s2 − 0.6)
3) and, work of the spring force.
US = - 0.5 (200) (s2−0.6)2 = -100 (s2 − 0.6)2
EXAMPLE (continued)
The work and energy equation will be
T1 + U1-2 = T2
0.5 (10) 52 + 100(s2 − 0.6) + 49.05(s2 − 0.6) − 100(s2 − 0.6)2 = 0
 125 + 149.05(s2 − 0.6) − 100(s2 − 0.6)2 = 0
Solving for (s2 − 0.6),
(s2 − 0.6) = {-149.05 ± (149.052 – 4×(-100)×125)0.5} / 2(-100)
Selecting the positive root, indicating a positive spring deflection,
(s2 − 0.6) = 2.09 m
Therefore, s2 = 2.69 m
GROUP PROBLEM SOLVING
Given: The 2 lb brick slides down a
smooth roof, with vA=5 ft/s.
Find:
The speed at B,
the distance d from the wall to
where the brick strikes the
ground, and its speed at C.
C
Plan:
1) Apply the principle of work and energy to the brick, and determine the
speeds at B and C.
2) Apply the kinematic relations in x and y-directions.
GROUP PROBLEM SOLVING (continued)
Solution:
Solving for the unknown velocity yields vB = 31.48 ft/s
vC = 54.1 ft/s
GROUP PROBLEM SOLVING (continued)
2) Apply the kinematic relations in x and y-directions:
Equation for horizontal motion
(+ ) xC = xB + vBx tBC
d = 0 + 31.48 (4/5) tBC
 d = 6.996 tBC
Equation for vertical motion
(+) yC = yB + vBy tBC – 0.5 g tBC2
 -30 = 0 + (-31.48)(3/5) tBC – 0.5 (32.2) tBC2
Solving for the positive tBC yields tBC = 0.899 s.
 d = 6.996 tBC = 6.996 (0.899) = 22.6 ft

C
ATTENTION QUIZ
1.
What is the work done by the normal force N if a 10
lb box is moved from A to B ?
A)
- 1.24 lb · ft
C) 1.24 lb · ft
2.
B) 0 lb · ft
D) 2.48 lb · ft
Two blocks are initially at rest. How many equations would be needed to
determine the velocity of block A after block B moves 4 m horizontally on the
smooth surface?
A)
One
B)
Two
C)
Three
D)
Four
2 kg
2 kg